Questions Further Paper 2 (305 questions)

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AQA Further Paper 2 2024 June Q9
4 marks Standard +0.8
A curve passes through the point \((-2, 4.73)\) and satisfies the differential equation $$\frac{dy}{dx} = \frac{y^2 - x^2}{2x + 3y}$$ Use Euler's step by step method once, and then the midpoint formula $$y_{r+1} = y_{r-1} + 2hf(x_r, y_r), \quad x_{r+1} = x_r + h$$ once, each with a step length of \(0.02\), to estimate the value of \(y\) when \(x = -1.96\) Give your answer to five significant figures. [4 marks]
AQA Further Paper 2 2024 June Q10
4 marks Standard +0.8
The matrix \(\mathbf{C}\) is defined by $$\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}$$ Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]
AQA Further Paper 2 2024 June Q11
3 marks Standard +0.8
Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots. Latifa says that if such an equation has exactly one real root, it must be of degree 3 Sam says that this is not correct. State, giving reasons, whether Latifa or Sam is right. [3 marks]
AQA Further Paper 2 2024 June Q12
5 marks Challenging +1.2
The transformation \(S\) is represented by the matrix \(\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}\) The transformation \(T\) is a reflection in the line \(y = x\sqrt{3}\) and is represented by the matrix \(\mathbf{N}\) The point \(P(x, y)\) is transformed first by \(S\), then by \(T\) The result of these transformations is the point \(Q(3, 8)\) Find the coordinates of \(P\) Give your answers to three decimal places. [5 marks]
AQA Further Paper 2 2024 June Q13
8 marks Standard +0.8
  1. Use the method of differences to show that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} = \frac{1}{4} - \frac{1}{2n} + \frac{1}{2(n + 1)}$$ [5 marks]
  2. Find the smallest integer \(n\) such that $$\sum_{r=2}^{n} \frac{1}{(r - 1)r(r + 1)} > 0.24999$$ [3 marks]
AQA Further Paper 2 2024 June Q14
10 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined as $$\mathbf{M} = \begin{bmatrix} 5 & 2 & 1 \\ 6 & 3 & 2k + 3 \\ 2 & 1 & 5 \end{bmatrix}$$ where \(k\) is a constant.
  1. Given that \(\mathbf{M}\) is a non-singular matrix, find \(\mathbf{M}^{-1}\) in terms of \(k\) [5 marks]
  2. State any restrictions on the value of \(k\) [1 mark]
  3. Using your answer to part (a), show that the solution to the set of simultaneous equations below is independent of the value of \(k\) \(5x + 2y + z = 1\) \(6x + 3y + (2k + 3)z = 4k + 3\) \(2x + y + 5z = 9\) [4 marks]
AQA Further Paper 2 2024 June Q15
7 marks Standard +0.8
The diagram shows the line \(y = 5 - x\) \includegraphics{figure_15}
  1. On the diagram above, sketch the graph of \(y = |x^2 - 4x|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) [3 marks]
  2. Find the solution of the inequality $$|x^2 - 4x| > 5 - x$$ Give your answer in an exact form. [4 marks]
AQA Further Paper 2 2024 June Q16
9 marks Challenging +1.2
The function f is defined by $$f(x) = \frac{ax + 5}{x + b}$$ where \(a\) and \(b\) are constants. The graph of \(y = f(x)\) has asymptotes \(x = -2\) and \(y = 3\)
  1. Write down the value of \(a\) and the value of \(b\) [2 marks]
  2. The diagram shows the graph of \(y = f(x)\) and its asymptotes. The shaded region \(R\) is enclosed by the graph of \(y = f(x)\), the \(x\)-axis and the \(y\)-axis. \includegraphics{figure_16}
    1. The shaded region \(R\) is rotated through \(360°\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [3 marks]
    2. The shaded region \(R\) is rotated through \(360°\) about the \(y\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [4 marks]
AQA Further Paper 2 2024 June Q17
9 marks Standard +0.8
The Argand diagram below shows a circle \(C\) \includegraphics{figure_17}
  1. Write down the equation of the locus of \(C\) in the form $$|z - w| = a$$ where \(w\) is a complex number whose real and imaginary parts are integers, and \(a\) is an integer. [2 marks]
  2. It is given that \(z_1\) is a complex number representing a point on \(C\). Of all the complex numbers which represent points on \(C\), \(z_1\) has the least argument.
    1. Find \(|z_1|\) Give your answer in an exact form. [3 marks]
    2. Show that \(\arg z_1 = \arcsin\left(\frac{6\sqrt{3} - 2}{13}\right)\) [4 marks]
AQA Further Paper 2 2024 June Q18
4 marks Standard +0.8
In this question you may use results from the formulae booklet without proof. Use the binomial series for \((1 + x)^n\) and the Maclaurin's series for \(\sin x\) to find the series expansion for \(\frac{1}{(1 + \sin \theta)^4}\) up to and including the term in \(\theta^3\) [4 marks]
AQA Further Paper 2 2024 June Q19
10 marks Challenging +1.2
Solve the differential equation $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} - 45y = 21e^{5x} - 0.3x + 27x^2$$ given that \(y = \frac{37}{225}\) and \(\frac{dy}{dx} = 0\) when \(x = 0\) [10 marks]
AQA Further Paper 2 2024 June Q20
9 marks Challenging +1.3
The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{\pi}{4}} \cos^n x \, dx \quad\quad (n \geq 0)$$
  1. Show that $$I_n = \left(\frac{n-1}{n}\right)I_{n-2} + \frac{1}{n\left(2^{\frac{n}{2}}\right)} \quad\quad (n \geq 2)$$ [6 marks]
  2. Use the result from part (a) to show that $$\int_0^{\frac{\pi}{4}} \cos^6 x \, dx = \frac{a\pi + b}{192}$$ where \(a\) and \(b\) are integers to be found. [3 marks]
AQA Further Paper 2 Specimen Q1
1 marks Easy -1.8
Given that \(z_1 = 4e^{i\frac{\pi}{3}}\) and \(z_2 = 2e^{i\frac{\pi}{4}}\) state the value of \(\arg\left(\frac{z_1}{z_2}\right)\) Circle your answer. [1 mark] \(\frac{\pi}{12}\) \quad \(\frac{4}{3}\) \quad \(\frac{7\pi}{12}\) \quad \(2\)
AQA Further Paper 2 Specimen Q2
3 marks Easy -1.2
Given that \(z\) is a complex number and that \(z^*\) is the complex conjugate of \(z\) prove that \(zz^* - |z|^2 = 0\) [3 marks]
AQA Further Paper 2 Specimen Q3
3 marks Standard +0.8
The transformation T is defined by the matrix M. The transformation S is defined by the matrix \(\mathbf{M}^{-1}\). Given that the point \((x, y)\) is invariant under transformation T, prove that \((x, y)\) is also an invariant point under transformation S. [3 marks]
AQA Further Paper 2 Specimen Q4
4 marks Standard +0.3
Solve the equation \(z^3 = i\), giving your answers in the form \(e^{i\theta}\), where \(-\pi < \theta \leq \pi\) [4 marks]
AQA Further Paper 2 Specimen Q5
4 marks Standard +0.3
Find the smallest value \(\theta\) of for which \((\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)\) \(\{\theta \in \mathbb{R} : \theta > 0\}\) [4 marks]
AQA Further Paper 2 Specimen Q6
5 marks Standard +0.3
Prove that \(8^n - 7n + 6\) is divisible by 7 for all integers \(n \geq 0\) [5 marks]
AQA Further Paper 2 Specimen Q7
5 marks Challenging +1.2
A small, hollow, plastic ball, of mass \(m\) kg is at rest at a point \(O\) on a polished horizontal surface. The ball is attached to two identical springs. The other ends of the springs are attached to the points \(P\) and \(Q\) which are 1.8 metres apart on a straight line through \(O\). The ball is struck so that it moves away from \(O\), towards \(P\) with a speed of 0.75 m s\(^{-1}\). As the ball moves, its displacement from \(O\) is \(x\) metres at time \(t\) seconds after the motion starts. The force that each of the springs applies to the ball is \(12.5mx\) newtons towards \(O\). The ball is to be modelled as a particle. The surface is assumed to be smooth and it is assumed that the forces applied to the ball by the springs are the only horizontal forces acting on the ball.
  1. Find the minimum distance of the ball from \(P\), in the subsequent motion. [5 marks]
AQA Further Paper 2 Specimen Q7
2 marks Moderate -0.5
  1. In practice the minimum distance predicted by the model is incorrect. Is the minimum distance predicted by the model likely to be too big or too small? Explain your answer with reference to the model. [2 marks]
AQA Further Paper 2 Specimen Q8
5 marks Standard +0.8
Given that \(I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx\) \quad \(n \geq 0\) show that \(n I_n = (n-1)I_{n-2}\) \quad \(n \geq 2\) [5 marks]
AQA Further Paper 2 Specimen Q9
6 marks Challenging +1.2
A student claims: "Given any two non-zero square matrices, A and B, then \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)"
  1. Explain why the student's claim is incorrect giving a counter example. [2 marks]
  2. Refine the student's claim to make it fully correct. [1 mark]
  3. Prove that your answer to part (b) is correct. [3 marks]
AQA Further Paper 2 Specimen Q10
8 marks Challenging +1.8
Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]
AQA Further Paper 2 Specimen Q11
8 marks Challenging +1.8
The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics{figure_11} The polar equation of \(C\) is \(r = 4 + 2\cos \theta\), \quad \(-\pi \leq \theta \leq \pi\)
  1. Show that the area of the region bounded by the curve \(C\) is \(18\pi\) [4 marks]
  2. Points \(A\) and \(B\) lie on the curve \(C\) such that \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) and \(AOB\) is an equilateral triangle. Find the polar equation of the line segment \(AB\) [4 marks]
AQA Further Paper 2 Specimen Q12
11 marks Standard +0.8
\(\mathbf{M} = \begin{pmatrix} -1 & 2 & -1 \\ 2 & 2 & -2 \\ -1 & -2 & -1 \end{pmatrix}\)
  1. Given that 4 is an eigenvalue of M, find a corresponding eigenvector. [3 marks]
  2. Given that \(\mathbf{MU} = \mathbf{UD}\), where D is a diagonal matrix, find possible matrices for D and U. [8 marks]